Ind. Eng. Chem. Res. 2004, 43, 6847-6854
6847
Free-Volume Theory Applied to Diffusion in Liquids: A Critical Analysis Guillain Mauviel and Eric Favre* Laboratoire des Sciences du Ge´ nie Chimique, ENSIC, 1 rue Grandville, BP 451-54001 Nancy Cedex, France
The aim of this paper is to evaluate free-volume theory when applied to diffusion in liquids. A classical strategy for computing diffusive mass transfer consists of using the Vignes equation, but this empirical interpolation is restricted to binary systems. Thus, particular emphasis has been placed on the development of a multicomponent model based on Vrentas and Duda approach, which is widely accepted for polymeric systems. For weakly interacting species, it is shown that robust mass-transfer computations can be achieved if local tracer diffusivities, which can be theoretically computed by free-volume theory, are known. A case study based on the benzenecyclohexane system has been performed and shows that the simplest free-volume formalism does not achieve a good prediction of experimental tracer diffusion data. Nevertheless, accurate interpolations are obtained when the excess volume is taken into account. Finally, the empirical Vignes relationship is shown to be consistent with basic free-volume arguments. Introduction The theoretical analysis and computation of diffusion phenomena in liquids is a problem of utmost importance for a great number of situations.1,2 This topic is complicated by the fact that, from both the theoretical and experimental points of view, different types of diffusion coefficients have to be considered: (i) Engineering computations most often make use of Fickian mutual diffusion coefficients of each species i, Dim, which can be experimentally determined under stationary or transitory mass-transfer conditions. When the molar convective reference velocity is chosen, the stationary specific flow rate of species i, Ni, is then expressed as n
x Ni ) - Dim C∇xi + xi
∑ Nk
(1)
k)1
x is the molar mutual Fickian diffusivity of species i. Dim C is the total concentration, and xi is the mole fraction of i. For a binary system (n ) 2), the Fickian diffusivities x x x ) D2m ) D12 , and do not are symmetrical, i.e., D1m depend of the choice of reference velocity (mole, mass, x ω Φ ) D12 ) D12 . or volume), i.e., D12 (ii) A second type of diffusion coefficient must also be considered when departure from conditions of thermodynamic ideality are taken into account. This approach, which intends to express the rigorous driving force for mass transfer (i.e., the chemical potential gradient instead of the sole concentration gradient postulated by Fick’s law), ends up with so-called thermodynamic diffusion coefficients Dim. Such coefficients are typically used in the generalized form of Fick’s law
x Ci Ni ) -Dim
∇µi RT
n
+ xi
∑ Nk
(2)
k)1
* To whom correspondence should be adressed. Tel.: +33 (0)3 83 17 52 84. Fax: +33 (0)3 83 32 29 75. E-mail:
[email protected].
Here, the diffusive flow rate of species i is a function of the gradient of its own of chemical potential; the influence of the other gradients is not considered as in classical irreversible thermodynamics.3 The binary diffusivities used in the Stefan-Maxwell equation can also reduce to these thermodynamic diffusivities under appropriate assumptions. For a binary system, the molar thermodynamic difx , are symmetrical. fusivities, Dim (iii) Specific techniques such as spin-echo NMR spectroscopy or radioactive labeling enable the tracer diffusion coefficient, Di, of a compound to be assessed, that is, the mobility of a given molecule in the absence of a concentration gradient. This tracer diffusion is often called self-diffusion. In our opinion, however, the latter terminology should be reserved to the diffusion of a given molecule in its pure liquid. An increasing number of experimental data of this type have been reported in the literature with the development of NMR techniques in research laboratories. The various diffusivities described above are often concentration dependent in a dense solution. Their typical behaviors are represented on Figure 1 for a binary system in the liquid phase. The differences between these coefficients vanish for infinitely dilute conditions. A single value, D°ik, is then obtained, which can be predicted using correlations (such as Wilke-Chang). A difficulty arises, however, as soon as nondilute conditions prevail. In that case, care has to be taken to consider the type of diffusion coefficient that is relevant. Generally speaking, equations enabling the correlation or prediction of the mutual diffusion coefficient from easily experimentally accessible data are of considerable importance for mass-transfer computations. The first target has been to predict the concentration dependency of the mutual diffusion coefficient from tracer diffusivities. For a binary system, a relationship was proposed to achieve that purpose by Hartley and Crank for liquids4 and by Darken for solids5
Dx12 ) x1D2 + x2D1
10.1021/ie049719w CCC: $27.50 © 2004 American Chemical Society Published on Web 09/17/2004
(3)
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Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004
Figure 1. Various types of diffusivities of a binary system and their common trends.
Experimental results6,7 show that this linear relation is valid for weakly interacting systems. Actually, eq 3 is demonstrated by Bearman’s statistical mechanical theory8 only if the rule of the geometric mean can be applied to the friction coefficients (ζ12 ) xζ11ζ22). To date, this equation has not been used for predictions because the concentration dependencies of tracer diffusivities are not well correlated. In contrast, diffusion coefficients under infinitely dilute conditions, D°ik, are easily accessible and/or well correlated. Thus, many authors have proposed empirical interpolations for computing mutual diffusion coefficients from these values. Caldwell and Babb9 proposed a linear interpolation. Leffler and Cullinan10 and Asfour and Dulien11 pointed out the influence of the viscosity of the solution. However, Vignes’ relation12 is by far the most often used
Dx12 ) (D°21)x1(D°12)x2
(4)
This logarithmic interpolation shows satisfactory predictive ability when it is applied to ideal or nearly ideal mixtures.12,13 Large deviations have been reported for systems that are subject to association. Nevertheless, the extension of this binary expression to multicomponent systems2 seems quite uncertain. Furthermore, the fact that Vignes’ expression is empirical is obviously a drawback. Thus, theoretical developments have been also attempted to achieve a similar purpose on an expectedly more robust basis. Activated diffusion theories have been suggested postulating that an equivalent reaction takes place when diffusion proceeds,14,15 with the intention of demonstrating the Vignes relation as a particular case. However, this theoretical approach did not lead to better predictions. The incidence of molecular deviations from random mixing situations has also been a subject of considerable interest for so-called cluster diffusion phenomena.16,17 It is striking however that, to our knowledge, a theoretical approach to the problem based on freevolume theory has never been attempted. Free-volume theory was originally developed by Cohen and Turnbull18 for the prediction of variations in the viscosity of pure liquids. This concept was later extended to the computation of tracer diffusion coef-
ficients in polymer-liquid systems by Fujita19 and Vrentas and Duda.20 Currently, the latter model is certainly the most widely accepted model for application to polymer solutions. Nevertheless, the scope of this theory has been limited to polymer diffusion, while its tentative applicability to more conventional liquid systems has not been explored. It is the intention of this work to investigate the implications and theoretical predictions of free-volume theory when it is applied to liquid mixtures. In a first step, the case of diffusion in multicomponent mixtures is treated. Possibilities for computing mass transfer from tracer diffusivities are presented. To predict the concentration dependency of these tracer diffusivities, the Vrentas-Duda model is formulated on a generic basis (polymeric or liquid systems). This formulation is then assessed on the benzene-cyclohexane mixture to check the relevance of free-volume theory for diffusion in simple liquids. Finally, it is shown that the Vignes relationship can be directly related to freevolume theory. Diffusion in Multicomponent Mixtures As mentioned in the Introduction, a binary system has a symmetrical mutual diffusion coefficient that does not depend on the choice of convective reference velocity. For multicomponent mixtures, this choice of reference does affect the mutual diffusion coefficient.1 The volume reference velocity leads to the expression n
Φ V h iNi ) -Dim ∇Φi + Φi
∑ Vh kNk )
k)1
Φ Φi -Dim
∇µi RT
n
+ Φi
∑ Vh kNk k)1
(5)
and the mass reference velocity to n
ω F∇ωi + ωi MiNi ) -Dim
∑ MkNk ) k)1 ω Fi -Dim
∇µi RT
n
+ ωi
∑ MkNk k)1
(6)
x If n > 2, the mutual diffusivities are not equal: Dim * Φ ω Dim * Dim. Generaly speaking, neither the Fickian nor the thermodynamic diffusion coefficients are independent of the system composition. Moreover, there are no empirical or theoretical relations to predict their concentration dependency. In contrast, tracer diffusivities are well correlated by free-volume theory in polymer-solvent systems. Thus, general mass-transfer computations could be achieved if this transfer could be related to the tracer diffusivities and if these tracer diffusivities could be predicted for any multicomponent system (polymeric or liquid). Here, it has to be stressed that Maxwell-Stefan diffusivities are also concentration dependent for dense solutions. The extension of the Vignes relation to multicomponent mixtures2 for computation of the Maxwell-Stefan diffusivities is uncertain.
Mass-Transfer Computations from Tracer Diffusivities To relate thermodynamic diffusivity and tracer diffusivity, equality between these coefficients has been postulated.
Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004 6849 x Specifically, Bitter21 proposed the equality D1 ) D1m for a solvent (1)-polymer (2) system. Because the molar thermodynamic diffusivity is symmetrical, this relation x ) D2. is obviously wrong: it would imply D1 ) D12 In contrast, the volume (or mass) thermodynamic diffusivities are not symmetrical. Indeed, the volume Fickian diffusivity is related to the volume thermodynamic diffusivities by
Dx12 ) D12Φ ) DΦ 1m
(
)
∂ ln a1 ∂ ln C1
) DΦ 2m
T,P
(
∂ ln a2 ∂ ln C2
)
Φ ) Di Dim
Free-Volume Theory
(7)
T,P
Thus, the Gibbs-Duhem relationship leads Φ Φ /V h 2 ) D2m /V h 1. D1m Then, the following equality can be assumed
to
(8)
This stringent relation can be demonstrated by Bearman’s8 statistical mechanical theory if the friction coefficients of one component i with all of the surrounding components are related by
ζik V hk
)
ζij V hj
∀k, j
n
Ni ) -DiCi
RT
+ xi
The analysis of viscosity and diffusion in dense liquid systems has been based on Cohen and Turnbull’s freevolume theory18 for several decades. In this theory, the solution is viewed as jumping units (small molecules or segment of large molecules) separated by free volume. A part of this empty volume (the hole free volume) is redistributed statistically by thermal fluctuations. Hence, the motion of a particular jumping unit is linked to (i) the probability that a fluctuation in local density will produce a hole of sufficient size and (ii) the probability that a jumping unit will have sufficient energy to overcome attractive forces. This simplified view of the diffusion mechanism leads to the following expression for the tracer diffusion of one species i
(9)
Bearman called mixtures that obey this hypothesis “regular solutions”. It has been used recently by Alsoy and Duda22 (case 1). According to this hypothesis, the ratio of tracer diffusivities in a binary mixture should be equal to the ratio of their molar volumes: D1/D2 ) h 1. McCall and Douglass7 checked this relation and V h 2/V concluded that it is approximately true for nonassociated mixtures. When the mass reference velocity is chosen, it is possible in the same manner to postulate equality between mass thermodynamic diffusivity and tracer ω ) Di. This is demonstrated by Beardiffusivity: Dim 8 man’s statistical mechanical theory if23 ζik/Mk ) ζij/Mj ∀k, j. This would lead to D1/D2 ) M2/M1 for a binary system. Equation 9 involves the geometrical mean relationship (ζ12 ) xζ11ζ22), but the reverse is not true. As stated in the Introduction, this geometrical mean relationship, which is less stringent than eq 9, leads to the Darken relationship for binary systems (eq 3). For multicomponent systems, Price and Romdhane24 and Yamamura25 demonstrated recently that it leads to
∇µi
It should be noted here that expression 10 has been shown25 to work well for the ternary liquid mixture of acetone, benzene, and methanol.26 Nevertheless, Price and Romdhane24 did not validate it for weakly interacting polymer-solvent systems. This might be due to the large difference in size between the molecules, which would not fit the geometrical mean conditions. A forthcoming paper will address this concern.
Nk
∑D
k)1 n
xk
∑D
k)1
k
(10)
k
This expression should allow for the computation of mass transfer in any multicomponent systems as long as the geometrical mean relationship holds, i.e., for weakly interacting systems. Nevertheless, expression 10 is useful only if tracer diffusivities can be computed from the local composition. This concentration dependency could be correlated and even predicted by free-volume theory.
Di )
D∞i
( ) ( )
Ei γV h /ij exp exp RT V hf
(11)
where D∞i is the infinite-temperature diffusivity of species i; Ei is its activation energy; and γ is an overlap factor, which is introduced because the same free volume is available to more than one jumping unit. V h /ij is the critical volume of a species i jumping unit. V h f is the average hole free volume per jumping unit. Free-volume theory was adapted by Fujita19 to describe diffusion in polymer-solvent systems. Although his formalism correlates diffusion data quite well, it has no predictive capabilities. Hence, the rigorous VrentasDuda20 formalism is preferred. For instance, the tracer diffusion of a solvent 1 in a polymer 2 is written as
( )
E1 × RT -(ω1V ˆ /1 + ω2ξV ˆ /2) exp K11 K12 ω1 (K21 - Tg,1 + T) + ω2 (K22 - Tg,2 + T) γ1 γ2
D1 ) D∞1 exp -
[
]
(12)
where V ˆ /i is the critical local hole free volume per mass of species i. ξ is the ratio of the molar volumes of the / / /V h 2j . Finally, the shared mass jumping units: ξ ) V h 1j free volume of one component is given by V ˆ f,i/γi ) (K1i/γi)(K2i - Tg,i + T). All of these parameters are not just adjustable to allow good empirical fits to data. They all have a physical significance within free-volume theory. Zielinsky and Duda27 suggest that it is possible to determine these different parameters on a predicitive basis. Actually, the most ambiguous parameter, ξ, is often adjusted on the basis of diffusion data. Nevertheless, this formalism presents a difficulty for a newcomer: the physical meaning of the exponential numerator is not clear. Indeed, the critical volume of the polymer, V h /2, should not influence the tracer diffu-
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sion of the solvent. A better picture is obtained when the definition of ξ is used to express the numerator of the second exponential in eq 12 / ω1V ˆ /1 + ω2ξV h /2 ) V h 1j
(
ω2 ω1 + M1j M2j
)
(13)
where Mkj is the molar weight of jumping unit k. Actually, this sum is part of the definition of the average hole free volume per jumping unit, V h f. For an ncomponent system, this average hole free volume can be written as
V hf )
n
1 n
ωk
∑M
k)1
∑
n
ωkV ˆ f,k or V hf )
k)1
∑ xkjVh f,k
(14)
k)1
Table 1. Solvent Free-Volume Parameters28 a
n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-dodecane benzene cyclohexane acetone chloroform water
V h /ij (cm3/mol)
K1i/γi [× 103 cm3/(g‚K)]
K2i - Tg,i (K)
D∞i b (× 108 m2/s)
83.6 97.7 111.7 125.8 139.9 154.0 182.2 70.4 84.8 54.8 60.9 19.3
2.41 1.96 1.83 1.52 1.35 1.22 1.05 1.51 3.02 1.86 0.71 2.18
-38.89 -41.08 -55.42 -51.98 -54.72 -55.14 -57.96 -94.32 -157.81 -53.33 -29.43 -152.29
3.11 3.50 3.43 3.67 5.01 5.22 6.13 4.47 2.01 3.6 4.07 8.55
a All of the jumping units are considered to be the whole molecule (Mij ) Mi). b The activation energy term is included as it is considered to be too small to be evaluated.
kj
where xij is the molar fraction of jumping units, which is defined by
xij )
Mi xi Mij
(15)
Mk
n
∑ xkM
k)1
Dii )
kj
Thus, the Vrentas-Duda expression can easily be extended to an n-component system
ωk
n
ln Di ) ln
D∞i
-
Ei
V h /ij
∑M
k)1
-
RT
n
kj
V ˆ f,k
∑ ωk γ k)1
k
or
(16) ln Di ) ln D∞i -
Ei RT
Actually, the ab initio calculation of solvent freevolume parameters is based on the validity of freevolume theory to describe the solvent self-diffusion coefficient, Dii. Indeed, the parameters K1i/γi, K2i Tg,i, D∞i , and even Ei are often regressed from Dullien’s equation predicting the dependence of self-diffusion on temperature
V h /ij
n
V h f,k
∑ xkj γ k)1
k
In this way, the use of ξ, which depends on two components, is avoided. Instead, it is replaced by the use of the molar weight of the jumping unit (which is equal to the molar weight of the molecule for small solvents). Nevertheless, all of the parameters that have been tabulated for decades can be used in these expressions (see Table 1). It has to be stressed that the mass-average and molaraverage expressions (eq 16) lead to slightly different predictions. The mass-average expression is commonly preferred, but it is not clear why the molar-average expression should be rejected. For mixtures of solvents, xij ) xi because the whole molecule is considered to be the jumping unit. Therefore, the molar-average expression is simpler to use and is preferred for this type of situation. Relevance of Free-Volume Theory for Simple Liquids Theoretically, both expressions 16 could be used to predict tracer diffusion in simple liquids.
0.124 ×10-16 V h c,i2/3RT ηiMiV ˆi
(17)
ˆi where V h c,i is the critical molar volume of species i. V and ηi are its specific volume and its viscosity, respectively, both of which are temperature-dependent properties. The constant 0.124 × 10-16 is in units of mol2/3. Thus, it is quite surprising that free-volume theory is not used to predict tracer diffusion in liquid mixtures. This lack of interest in free-volume theory might be due to the fact that mixtures of small molecules are typically far above their glass transition temperature. In this case, the free volume is quite large, and diffusion is often believed to be controlled as well by a concentrationdependent energy of activation.14,15 Because it is not feasible to model both the variation of Ei and the variation of the free volume with composition, Ei is considered to be a constant in free-volume theory. This constant is computed for the pure solvent, but in general, it is not even possible to determine an accurate value so Ei is neglected. Then, it would be worth evaluating whether tracer diffusion in simple liquids could be correctly predicted by free-volume theory without the contribution of activation energy. The correct description of concentration dependency is assessed afterwards. For illustrative purpose, the benzene (1)-cyclohexane (2) case has been explored. When the free-volume data of Table 1 are used, the ab initio prediction of the cyclohexane tracer diffusivity at 25 °C is very poor (see Figure 2): the cyclohexane self-diffusion is overestimated (by more than 30%), and its tracer diffusivity is predicted to decrease with benzene content in contrast to the actual rise observed by Mills.29 Tracer diffusivities in ternary mixtures of acetone, benzene, and methanol were measured at 25 °C by Zielinski and Hanley.30 The ab initio predictions obtained with the free-volume parameters of Table 1 are compared to their experimental data in Figure 3. Very large discrepancies are observed.
Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004 6851
Figure 2. Cyclohexane tracer diffusivity in a benzene-cyclohexane mixture: Experiments29 and ab initio predictions by free-volume theory.
Figure 3. Tracer diffusivities (×105 cm2/s) in acetone-benzenemethanol mixtures: Experiments30 and ab initio predictions by free-volume theory. Table 2. Solvent Free-Volume Parameters Tabulated by Hong28 and Computed from Other Density and Viscosity Data K1i/γi [× 103 cm3/(g‚K)] benzene Hong28 this work cyclohexane Hong28 this work
K2i - Tg,i (K)
D∞i a (× 108 m2/s)
1.51 1.24
-94.32 -91.18
4.47 7.71
3.02 0.64
-157.81 -5.48
2.01 30.89
a The activation energy term is included as it is considered to be too small to be evaluated.
These incorrect ab initio predictions might be due to the computation of the solvent free-volume parameters from the Dullien equation (eq 17). To check this hypothesis, the nonlinear regression of K1i/γi, K2i - Tg,i, and D∞i was performed for benzene and cyclohexane using other tabulated viscosity31 and density32 data. The new set of free-volume parameters obtained are compared to Hong’s values in Table 2. Large discrepancies are observed, especially for cyclohexane. The cyclohexane tracer diffusivity predicted using these new parameters is better: the self-diffusion is correct, and the trend is right, but large quantitative errors are still observed (see Figure 2). The discrepancies in the parameter values can be explained by a lack of accurate viscosity data and an excessive sensitivity of the parameters to these data. Regardless, it is obvious that this predictive approach to finding free-volume parameters is not reliable. However, this trouble does not definitively prove that freevolume theory is not valid for liquid mixtures.
Figure 4. Benzene tracer diffusivity in a benzene-cyclohexane mixture: Experiments29 and interpolations by free-volume theory.
Figure 5. Cyclohexane tracer diffusivity in a benzene-cyclohexane mixture: Experiments29 and interpolations by free-volume theory. Table 3. Cyclohexane Shared Free Volume V h f,2/γ2 (cm3/mol) and Infinite-Temperature Diffusivity D∞i (m2/s) Computed from Limiting Diffusivities
benzene cyclohexane
V h f,2/γ2
D∞i × 108
22.64 21.65
4.21 7.13
One way to settle this concern would be to use infinite-dilution diffusivities, D°ik. If the D°ik values are computed by accurate correlations (Wilke-Chang, Scheibel, Reddy-Doraiwamy, Tyn and Calus, etc.33), the predictive power is kept. However, it is also possible to use experimental values of Dii and D°ik to compute freevolume parameters. An interpolation of tracer diffusivity is thus achieved instead of a prediction. For benzene-cyclohexane, tracer diffusivities have been interpolated on a free-volume basis. The benzene shared free volume was computed at 25 °C from Hong’s free-volume parameters (V h f,1/γ1 ) 24.02 cm3/mol). The infinite-temperature diffusivity, D∞i , of each species i and the cyclohexane shared free volume were adjusted on the basis of two limiting diffusivities (Dii and D°ik) (see Table 3). From Figures 4 and 5, it appears that free-volume theory is not able to describe the actual maximum of the tracer diffusivity. This would imply that free-volume theory is not valid for simple liquids. However, we propose the excess volume to be taken into account. Indeed, it seems logical to add the shared excess volume to the shared average hole free volume per jumping unit
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Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004
V hf V h f,1 V h f,2 V hE + x2 + ) x1 γ γ1 γ2 γ
(18)
This hypothesis is based on the idea that this excess volume contributes only to free volume. For benzene-cyclohexane, the excess volume has been measured34 accurately at 25 °C and correlated by the expression
V h E (cm3/mol) ) x1(1 - x1)[2.5564 0.577(2x1 - 1) + 0.0267(2x1 - 1) ] Its maximum value is 0.647 cm3/mol at x1 ) 0.44. The value of the overlap factor γ is needed, but it is not known precisely (0.5 < γ < 0.9). At its minimum value (0.5), the theoretical interpolations are very close to the experimental measurements, with a relative error of less than 4% (see Figures 4 and 5). These results would tend to suggest that free-volume theory without the contribution of the activation energy is not inevitably irrelevant for liquid mixtures. However, there is a regrettable lack of accurate measurements to extend the validity of the proposed excess volume correction. Indeed, it would be very interesting to evaluate whether the overlap factor γ is a constant or just another adjustable parameter. Regardless, the predictive power of free-volume theory seems to be lost with the need for the excess volume data. This would be an important limit for the use of such a model. Link to the Vignes Relationship
V h f,2 x1 + (1 - x1) γ2 γ1 V h f,1
(19)
The activation energy term is included in the infinitediffusion coefficient, D∞i . When x1 ≈ 0, the infinite-dilution diffusivity of 1 in 2 is
ln D°12 ) ln D∞1 -
/ V h 1j
V h f,2
V h f,i/γi
species
V h f,i/γi
24.0 35.6 26.4 22.8 5.7 32.6 39.6 43.2
n-C5H12 n-C6H14 n-C7H16 n-C8H18 n-C9H20 n-C10H22 n-C12H24
45.1 43.4 44.5 42.7 42.1 42.2 42.9
Calculated at 25 °C with Hong’s data.28
These two limiting cases allow eq 19 to be expressed in a different way
x1 ln D1 )
(20)
X1 )
V h f,1 γ1
(22)
x1G12 1 - x1(1 - G12)
(23)
with G12 is the ratio of the shared molar free volumes, i.e.
G12 )
V h f,1/γ1 V h f,2/γ2
D1 ) (D11)X1(D°12)1-X1
(24)
in which only shared molar free volume is needed. Some values at 25 °C are reported in Table 4. At this stage, a link between this binary formulation of free-volume theory and the Vignes empirical relationship can be made. Starting from eq 8, the self-diffusion of pure 1 is expressed as a function of the infinite-dilution diffusivity h 2/V h 1)D°21. By free-volume theory of 2 in 1: D11 ) (V (eq 24), the volume thermodynamic diffusivity is then
DΦ 1m )
( ) V h2 D°21 V h1
X1
(D°12)1-X1
(25)
For a binary system without excess volume, the relation between the volume and molar thermodynamic diffusivities is
DΦ 1m )
When x1 ) 1, the self-diffusion coefficient of pure 1 is / V h 1j
V h f,2 x1 + (1 - x1) γ2 γ1 V h f,1
At this stage, it is worth defining a modified molar fraction
γ2
ln D11 ) ln D∞1 -
V h f,2 V h f,1 ln D11 + (1 - x1) ln D°12 γ1 γ2
Thus, eq 22 becomes
In this section, we intend to show that the Vignes empirical interpolation can be related to free-volume theory. For a binary system of small molecules without excess volume, the tracer diffusivity of species 1 is given by / V h 1j
species benzene cyclohexane acetone chloroform water C2H6 C3H8 n-C4H10 a
2
ln D1 ) ln D∞1 -
Table 4. Shared Molar Free Volumes V h f,i/γi (cm3/mol) of Different Speciesa
V h2
Dx12 x1 V h 1 + (1 - x1)V h2
(26)
Because this molar thermodynamic diffusivity is symmetrical, it is more often used as in Vignes relationship. Our binary expression then becomes
(21) Dx12
[ ( )]( )
) 1 + x1
V h1
V h2
-1
V h2
V h1
X1
(D°21)X1(D°12)(1-X1) (27)
Ind. Eng. Chem. Res., Vol. 43, No. 21, 2004 6853
Figure 6. Ratio of the Vignes equation to eq 27 computed for different binary systems.
Figure 7. Molar thermodynamic diffusivity of the benzenecyclohexane system: Experiments29 and predictions.
This interpolation based on free-volume theory can be compared to Vignes’ eq 4. In the Figure 6, the ratio of eq 4 to eq 27 is represented for different regular solutions. This ratio appears to be very close to unity. Thus, the simple formalism of Vignes’ equation leads to predictions similar to those obtained with a theoretical development based on free-volume theory. Because the correction provided by eq 27 is small, there seems to be no reason to prefer this equation to the compact formulation of Vignes. Nevertheless, our theoretical approach opens the possibility for handling multicomponent systems, and it could be interesting for binary cases in which Vignes expression is not efficient. For instance, it is possible to predict the molar thermodynamic diffusivity of the system benzene-cyclohexane by using the Darken equation (eq 3) and the predictions of both tracer diffusivities by the free-volume expression 16 accounting for excess volume. In Figure 7, we see that this leads to minor errors compared to the Vignes prediction. Conclusions and Perspectives Diffusive mass transfer in liquid mixtures is currently predicted by computing mutual diffusivity through empirical relations, such as the Vignes logarithmic interpolation. Nevertheless, the extension of these binary relations to multicomponent mixtures seems doubtful. Robust predictions could be achieved if accurate models for computing tracer diffusivities and, from them, mass transfer were developed. For weakly interacting species, the friction coefficients are correctly linked by the geometrical mean relationship. This leads to the Darken expression for binary systems, but also to a new multicomponent expression using tracer diffusivities instead of mutual diffusivities.
Unlike these mutual diffusivities, tracer diffusivities can be computed from the local composition by freevolume theory. A formulation of Vrentas-Duda model has been proposed in this work that facilitates the extension to any multicomponent mixture. Even though free-volume theory is widely accepted for polymer-solvent mixtures, this is not the case for liquid systems far above glass transition temperature. Indeed, diffusion is then believed to be controlled as well by a concentration-dependent energy of activation. For the benzene-cyclohexane system, it was possible to obtain accurate interpolations by adding the shared excess volume to the shared average hole free volume. However, there is a regrettable lack of accurate measurements that could be used to extend the validity of the proposed excess volume correction. Moreover, this correction is limited in its utility by the fact that excess volume is difficult to predict. Nevertheless, it appears that free-volume theory without the activation energy contribution is not inevitably irrelevant for liquid mixtures. Thus, this approach opens the possibility for dealing with multicomponent systems or with binary systems for which the Vignes expression is not efficient. Moreover, Vignes’ equation was compared to a theoretical expression built on free-volume theory and a proportionality relationship between the friction coefficients. Their similar predictions provides a stronger consistency to the Vignes empiricial interpolation, which remains a cornerstone in diffusion coefficient computations for binary systems. If free-volume theory appears to be a promising approach for predicting diffusive mass transfer in any system, it has to be highlighted that the actual databank is subject to questions. Beyond the fact that data for many compounds are not yet tabulated, it seems that some values lead to inaccurate predictions. Particularly, it appears that the computation of solvent parameters through the Dullien equation is not reliable. Obviously, this computation based on solvent self-diffusion would be improper if free-volume theory were proven to be irrelevant for simple liquids. Regardless, it would be more judicious to compute the solvent parameters by using experimental or correlated diffusion data. Acknowledgment Special thanks are due to J. Roman Galdamez and Professor J. L. Duda for their helpful remarks. Nomenclature C ) concentration (mol‚m-3) Dim ) fickian mutual diffusivity of species i (m2‚s-1) Dim ) thermodynamic mutual diffusivity of species i (m2‚s-1) Di ) tracer diffusivity of species i (m2‚s-1) Dii ) self-diffusion of species i (m2‚s-1) D°ik ) infinite-dilution diffusivity of species i in k (m2‚s-1) D∞i ) infinite-temperature diffusivity of species i (m2‚s-1) Ei ) jumping activation energy of species i (J‚mol-1) K1i/γi ) shared free-volume parameter of species i (m3‚kg-1‚K-1) K2i - Tg,i ) shared free-volume parameter of species i (K) Mi ) molecular weight of species i (kg‚mol-1) Ni ) specific molar flow rate of species i (mol‚m-2‚s-1) n ) number of species R ) gas constant () 8.31441 J‚mol-1‚K-1) T ) temperature (K)
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V h i ) partial molar volume of species i (m3‚mol-1) V h E ) excess molar volume (m3‚mol-1) V ˆ i ) specific volume of species i (m3‚kg-1) V h /ij ) molar critical volume of the juming unit of species i (m3‚mol-1) V h f,i ) molar free volume of species i (m3‚mol-1) V ˆ f,i ) specific free volume of species i (m3‚kg-1) V h c,i ) critical molar volume of species i (m3‚mol-1) xi ) mole fraction of species i Greek Characters γ ) overlap factor ηi ) viscosity of species i (Pa‚s) Φi ) volume fraction of species i µi ) chemical potential of species i (J‚mol-1) ωi ) mass fraction of species i F ) density (kg‚m-3) ζik ) friction coefficient between species i and k (J‚m‚s‚mol-2)
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Received for review April 8, 2004 Revised manuscript received July 23, 2004 Accepted August 11, 2004 IE049719W
